"Setbacks are unavoidable. Giving up is unforgiveable," Biden said at the White House Rose Garden as he addressed staff who were disappointed in Vice President Kamala Harris' defeat.
"A defeat does not mean we are defeated."
Biden said Tuesday's election had proven the integrity of the US electoral system and said he would preside over an orderly transfer of power - an implicit rebuke of Trump, who sought to overturn his 2020 defeat to Biden and raised claims of fraud this year as well.
"We lost this battle. The America of your dreams is calling for you to get back up," he said.
Join me as I address the nation to discuss the election results and the transition. — President Biden (@POTUS) https://t.co/9LQhawXbxXNovember 7, 2024
Some Democrats have blamed Biden, 81, for Harris' defeat, saying he should not have sought re-election.
Biden only dropped his re-election bid in July after a disastrous TV debate with Trump raised alarm bells about his mental fitness.
Trump's campaign said Biden had invited him to meet at the White House at an unspecified time.
In the weeks ahead, Trump will select personnel to serve under his leadership.
Harris sought on Wednesday to console supporters.
Like Biden, she promised to aid Trump's transition before his inauguration on January 20 but urged Democrats to continue to fight for what they believe in.
Some Democrats worried their loss in Tuesday's presidential election showed that their values - progressive, socially liberal - were now firmly a minority among Americans in a divisive campaign.
Others were frustrated with the party's leadership, who they said had lost touch with much of the electorate who wanted help with the rising cost of living.
Trump's victory, surprisingly decisive after opinion polls that had shown a neck-and-neck contest, underscored how disenchanted many voters had become with the economy - in particular inflation - along with border security and the direction of the country and its culture.
Hispanics, traditionally Democratic voters, and lower-income households hit hardest by inflation helped fuel Trump's victory.
Republicans wrested the US Senate from Democrats in Tuesday's vote, ensuring Trump's party will control at least one chamber of Congress next year.
Republicans also held an edge in the battle for the House of Representatives although 38 of the 435 races still had no winner.
Trump prevailed in five of the seven battleground states to give him at least 295 Electoral College votes, more than the 270 needed to win the presidency.
He was leading in the remaining two, Arizona and Nevada, where votes were still being tallied.
Trump was also on track to become the first Republican presidential candidate to win the popular vote since George W Bush two decades ago.
Among people who may figure in Trump's leadership, Tesla CEO Elon Musk, the world's richest man and a prominent Trump donor, has been promised a role in his administration, as has former presidential candidate Robert F Kennedy Jr.
Ukrainian President Volodymyr Zelenskiy was among world leaders congratulating Trump.
But Trump has been critical of Biden's assistance for Ukraine in its war with Russia.
He has said he could end the war in 24 hours but has not offered a detailed plan.
The White House plans to rush billions of dollars in security assistance to Ukraine before Biden leaves office in January, sources said on Wednesday, hoping to shore up the government in Kyiv before Trump takes over.
